Question 1: Plants A, B, and C produce products I, II, and III. The rate of production (units per hour) of each product in the different plants is given in the table below: PRODUCT PLANT A B C I 4 6 5 II 3 4 3 III 2 3 2 The three plants may purchase a limited number of production hours at different prices according to the following table: PLANT MAXIMUM NUMBER OF HOURS PRICE PER HOUR A 250 $ 10 B 300 $ 8 C 200 $ 12 The demand and the unit sale price for the 3 products are given in the following table: PRODUCT DEMAND PRICE PER UNIT I 500 units $ 50 II 800 units $ 40 III 700 units $ 30 The demand for each product must be met. Any additional unit produced is sold at the same price. The goal is to maximize the total profit. 1. Formulate as an LP model. 2. Implement your model in Python and solve it using Gurobi Solver. What is the optimal production policy and what is the profit under this policy? (Provide a screenshot or PDF version of your Python Model and Solution) [20 Marks for Question 1; 10 for a correct formulation and 10 for the Python model] Question 2: You have solved a minimization problem with 3 variables and 2 constraints and have printed out the following sensitivity report: Microsoft Excel Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$4 Value: X1 0 3 5 1E+30 3 $D$4 Value: X2 0 1 3 1E+30 1 $E$4 Value: X3 1 0 4 2 4 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $F$8 Constraint1: 2 2 2 1E+30 1 $F$9 Constraint2: 2 0 1 1 1E+30 Answer the following questions: a. Does the solution have multiple optimal solutions? Provide a short explanation. b. What is the optimal objective function value if the RHS value of the first constraint increases to 7? Provide a short explanation. c. What is the optimal objective function value if the RHS value of the first constraint decreases to 1? Provide a short explanation. d. What is the optimal objective function value if the RHS value of the second constraint increases to 3? Provide a short explanation. e. Will the current solution remain optimal if the objective function coefficients for X1 and X3 both decrease by 1? Provide a short explanation. [25 Marks for Question 2; 5 for each section] Question 3: A company produces and sells three products (A, B and C). the operations manager is tasked with planning the level of production for the next 4 months. It is assumed that any number of products produced can be sold, generating the following per unit profit (differences are due to seasonality of the products): PRODUCT A PRODUCT B PRODUCT C Month 1 $ 140 $ 155 $ 210 Month 2 $ 140 $ 140 $ 220 Month 3 $ 150 $ 150 $ 230 Month 4 $ 150 $ 155 $ 230 Two resources are required to produce each product, labour hours and raw materials. The following table provide the number of units of each resource required to produce each product. PRODUCT A PRODUCT B PRODUCT C LABOUR (HOURS) 2 3 4 RAW MATERIAL (KG) 4 3 5 The company has already placed future orders for raw material for the next 4 months. These orders cannot be updated. You can assume that any amount of raw material that is not used in a given month can be carried to and used in the next month(s) without incurring any holding cost. For example, if you receive 100 kilograms of raw material in Month 1 and choose to only use 90 kilograms for production, 10 kilograms are added to your raw material availability in Month 2. The company has planned its workforce and has a certain number of available labour hours for each of the next 4 months. The differences between months are due to planned annual leave of its employees. In order to increase productivity, a survey among workers has been undertaken and the company has determined the number of overtime it will be able to use each month. Clearly, the company can decide not to have the employees work over time even if they are willing to. Furthermore, unused labour cannot be used in the future (for example, if you have 100 hours of labour in Month 1, but only choose to use 90 you cannot “store” the 10 unused hours for future use.) Due to the fact that the company’s employees earn different salaries, the hourly cost of overtime will vary over time. The relevant information is provided in the table below: The goal of the company is clearly to maximize the total 4-month profit. 1. Formulate as an LP model. 2. Implement your model in Excel and solve it using Solver. Provide a screenshot or PDF version of your model with the optimal solution. Provide an answer report. 3. Provide a complete policy suggestion based on your results. Include any information that may be relevant to your CEO. [55 Marks for Question 3; 35 for the correct LP formulation; 10 for an appropriate Excel Model and solution; 10 for Section 3] RAW MATERIAL ORDERED (KG) AMOUNT OF REGULAR LABOUR AVAILABLE (HOURS AMOUNT OF OVERTIME AVAILABLE (HOURS) OVERTIME COST PER HOUR Month 1 120 100 30 $ 25.0 Month 2 140 80 20 $ 25.0 Month 3 150 90 30 $ 28.0 Month 4 110 100 10 $ 30.0
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